Elegant integral property true?

Question

Let $f$ be a positive continuous function that decays sufficiently fast at infinity.

Now, fix $c \in \mathbb R$.

Is it true that if $\int_{x:|x-c|>a}(x-c)f(x)\le 0$ for all $a\ge 0$, then this implies that $$ \int_{-\infty}^{\infty} (x-c)^3f(x)dx\le 0? $$

Please let me know if you have any questions.

Answer 1

Write $F(r) = \int_{x:|x-c|>r} (x-c) f(x) \, \mathrm{d}x$. Then by the Fubini's theorem,

\begin{align*} \int_{-\infty}^{\infty} (x-c)^3 f(x) \, \mathrm{d}x &= \int_{-\infty}^{\infty} \left( \int_{0}^{|x-c|} 2r \, \mathrm{d}r \right) (x-c) f(x) \, \mathrm{d}x \\ &= \int_{0}^{\infty} \int_{x:|x-c|>r}2r (x-c) f(x) \, \mathrm{d}x\mathrm{d}r \\ &= \int_{0}^{\infty} 2rF(r) \, \mathrm{d}r. \end{align*}

This is non-positive because $2rF(r) \leq 0$ for all $r \geq 0$.